The 4^th term of GP is 500 and its common ratio is 1/m , m $\in ℕ$ . Let S_n denote the sum of the first $n$ terms of this GP. If S₆ > S₅ + 1 and S₇ < S₆ + 1/2. then the number of possible values of m is_____________

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answer is 12.

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Detailed Solution

$\begin{array}{l} S_{6} - S_{5} > 1 \Rightarrow {ar}^{5} > 1 \\ S_{7} - S_{6} < \frac{1}{2} \Rightarrow {ar}^{6} < \frac{1}{2} \end{array}$
Also given ar³ = 500

$\Rightarrow 500 r^{2} > 1 \Rightarrow r^{2} > \frac{1}{500}$

$r = 1 / m$ $\Rightarrow m^{2} < 500$
$r^{3} < \frac{1}{1000} \Rightarrow m > 10$

$m = 11, 12, 13 . . . 22$
No.of possible values of m are 12

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